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**Interpolation among reduced-order matrices to obtain parameterized models for design, optimization and probabilistic analysis.**
*(English)*
Zbl 1188.65110

Summary: Model reduction has significant potential in design, optimization and probabilistic analysis applications, but including the parameter dependence in the reduced-order model (ROM) remains challenging. In this work, interpolation among reduced-order matrices is proposed as a means to obtain parameterized ROMs. These ROMs are fast to evaluate and solve, and can be constructed without reference to the original full-order model. Spline interpolation of the reduced-order system matrices in the original space and in the space tangent to the Riemannian manifold is compared with Kriging interpolation of the predicted outputs. A heuristic criterion to select the most appropriate interpolation space is proposed. The interpolation approach is applied to a steady-state thermal design problem and probabilistic analysis via Monte Carlo simulation of an unsteady contaminant transport problem.

### MSC:

65L80 | Numerical methods for differential-algebraic equations |

35K20 | Initial-boundary value problems for second-order parabolic equations |

34A09 | Implicit ordinary differential equations, differential-algebraic equations |

65D05 | Numerical interpolation |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

65C05 | Monte Carlo methods |

### Keywords:

interpolation; parameterized reduced-order models; spline; Riemannian manifold; Kriging; differential-algebraic equations; numerical examples; convection-diffusion equation; steady-state thermal design problem; Monte Carlo simulation; unsteady contaminant transport problem
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\textit{J. Degroote} et al., Int. J. Numer. Methods Fluids 63, No. 2, 207--230 (2010; Zbl 1188.65110)

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